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Theorem advlog 24400

Description: The antiderivative of the logarithm. (Contributed by Mario Carneiro, 21-May-2016.)

**Assertion**
Ref	Expression
advlog	⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

**Proof of Theorem advlog**
Step	Hyp	Ref	Expression
1		reelprrecn 10028	. . . . 5 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. . . 4 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
3		rpre 11839	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
4	3	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
5	4	recnd 10068	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
6		1cnd 10056	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 1 ∈ ℂ)
7		recn 10026	. . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
8	7	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
9		1red 10055	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℝ)
10	2	dvmptid 23720	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 𝑥)) = (𝑥 ∈ ℝ ↦ 1))
11		rpssre 11843	. . . . . 6 ⊢ ℝ⁺ ⊆ ℝ
12	11	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ⊆ ℝ)
13		eqid 2622	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
14	13	tgioo2 22606	. . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
15		ioorp 12251	. . . . . . 7 ⊢ (0(,)+∞) = ℝ⁺
16		iooretop 22569	. . . . . . 7 ⊢ (0(,)+∞) ∈ (topGen‘ran (,))
17	15, 16	eqeltrri 2698	. . . . . 6 ⊢ ℝ⁺ ∈ (topGen‘ran (,))
18	17	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ (topGen‘ran (,)))
19	2, 8, 9, 10, 12, 14, 13, 18	dvmptres 23726	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ 1))
20		relogcl 24322	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
21	20	adantl 482	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
22		peano2rem 10348	. . . . . 6 ⊢ ((log‘𝑥) ∈ ℝ → ((log‘𝑥) − 1) ∈ ℝ)
23	21, 22	syl 17	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℝ)
24	23	recnd 10068	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − 1) ∈ ℂ)
25		rpreccl 11857	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1 / 𝑥) ∈ ℝ⁺)
26	25	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℝ⁺)
27	26	rpcnd 11874	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑥) ∈ ℂ)
28	21	recnd 10068	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29		dvrelog 24383	. . . . . . 7 ⊢ (ℝ D (log ↾ ℝ⁺)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥))
30		relogf1o 24313	. . . . . . . . . . 11 ⊢ (log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ
31		f1of 6137	. . . . . . . . . . 11 ⊢ ((log ↾ ℝ⁺):ℝ⁺–1-1-onto→ℝ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
32	30, 31	mp1i 13	. . . . . . . . . 10 ⊢ (⊤ → (log ↾ ℝ⁺):ℝ⁺⟶ℝ)
33	32	feqmptd 6249	. . . . . . . . 9 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)))
34		fvres 6207	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((log ↾ ℝ⁺)‘𝑥) = (log‘𝑥))
35	34	mpteq2ia 4740	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log ↾ ℝ⁺)‘𝑥)) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))
36	33, 35	syl6eq 2672	. . . . . . . 8 ⊢ (⊤ → (log ↾ ℝ⁺) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
37	36	oveq2d 6666	. . . . . . 7 ⊢ (⊤ → (ℝ D (log ↾ ℝ⁺)) = (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))))
38	29, 37	syl5reqr 2671	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
39		0cnd 10033	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 0 ∈ ℂ)
40		1cnd 10056	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 1 ∈ ℂ)
41		0cnd 10033	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → 0 ∈ ℂ)
42		1cnd 10056	. . . . . . . 8 ⊢ (⊤ → 1 ∈ ℂ)
43	2, 42	dvmptc 23721	. . . . . . 7 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ 1)) = (𝑥 ∈ ℝ ↦ 0))
44	2, 40, 41, 43, 12, 14, 13, 18	dvmptres 23726	. . . . . 6 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ 1)) = (𝑥 ∈ ℝ⁺ ↦ 0))
45	2, 28, 27, 38, 6, 39, 44	dvmptsub 23730	. . . . 5 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)))
46	27	subid1d 10381	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) − 0) = (1 / 𝑥))
47	46	mpteq2dva 4744	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 / 𝑥) − 0)) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
48	45, 47	eqtrd 2656	. . . 4 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − 1))) = (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑥)))
49	2, 5, 6, 19, 24, 27, 48	dvmptmul 23724	. . 3 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))))
50	24	mulid2d 10058	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1 · ((log‘𝑥) − 1)) = ((log‘𝑥) − 1))
51		rpne0 11848	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
52	51	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
53	5, 52	recid2d 10797	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 / 𝑥) · 𝑥) = 1)
54	50, 53	oveq12d 6668	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (((log‘𝑥) − 1) + 1))
55		ax-1cn 9994	. . . . . 6 ⊢ 1 ∈ ℂ
56		npcan 10290	. . . . . 6 ⊢ (((log‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
57	28, 55, 56	sylancl 694	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((log‘𝑥) − 1) + 1) = (log‘𝑥))
58	54, 57	eqtrd 2656	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥)) = (log‘𝑥))
59	58	mpteq2dva 4744	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((1 · ((log‘𝑥) − 1)) + ((1 / 𝑥) · 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
60	49, 59	eqtrd 2656	. 2 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥)))
61	60	trud 1493	1 ⊢ (ℝ D (𝑥 ∈ ℝ⁺ ↦ (𝑥 · ((log‘𝑥) − 1)))) = (𝑥 ∈ ℝ⁺ ↦ (log‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 {cpr 4179 ↦ cmpt 4729 ran crn 5115 ↾ cres 5116 ⟶wf 5884 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 +∞cpnf 10071 − cmin 10266 / cdiv 10684 ℝ⁺crp 11832 (,)cioo 12175 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 D cdv 23627 logclog 24301

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303

This theorem is referenced by: logfacbnd3 24948

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